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Abstract

The representation of geographic entities is characterized by inherent granularity due to scale and resolution specific observations. This article discusses the various aspects of rough set-based approximation modeling of spatial and conceptual granularity. It outlines the context and applications of rough set theory in representing objects with intermediate boundaries, spatial reasoning and knowledge discovery.

Granulation In Spatial Information

In geographic data information granulation is a central aspect of observation or abstract representation of geographic entities. The representation of space and spatial relationships in terms of idealized abstraction of geometric primitives (e.g., polygons, lines and points or pixels) and topological constraints inherently presupposes granulation of the space-time continuum. Spatial categories essentially represent hierarchies of abstractions. In spatial classification, the thematic classification systems are fixed by a well-defined crisp boundary or abstract concept in the context in which the data set was generated at a particular hierarchy. By establishing a coarse view of the world, it is possible to deal with large information granules such as states, districts, and counties. For example, in raster data model a regular grid is used to cover space where each grid cell uniformly represents spatial characteristics of local phenomena. Since the spatial variations of the local features within a grid are indistinguishable, a granular representation is implicit in a raster data model. The functional aspects of information processing also vary with the degrees of granularity. At finer details, the image processing tasks involve image segmentation, edge detection, and noise removal. While at higher end abstractions, computing algorithms are concerned with automated feature recognition, semantic interpretation, etc. The effect of granularity is also reflected in geostatistical prediction, which is classically known as the modifiable areal unit problem - a geographic manifestation of ecological fallacy causing spurious conclusion due to spatial aggregation (Openshaw, 1984).

In principle, granularity in an observation makes objects indiscernible, i.e., indistinguishable from each other. The indiscernibility relation can lead to the equivalence relation, which implies that objects within an equivalence class are reflexive (i.e., indistinguishable from itself), symmetric (i.e., if object x is indiscernible from y, then y is indiscernible from x), and transitive (if x is indiscernible from y and y is indiscernible from z then x is indiscernible from z). The notion of indiscernibility and the resulting equivalence class allows expressing concepts in terms of approximation space. However, the transitivity constraint may not always be maintained in spatial relations, for example, when the discernibility relation is a distance measure. Hence in those cases one may have to use the similarity or tolerance relation (i.e., objects are reflexive and symmetric).

Indiscernibility: A relationship among a finite collection of elements where objects are indiscernible with respect to a particular observation if any pair of elements in the collection cannot be distinguished from each other by the observation.

Granular Computing: Theories, tools and techniques that make use of granules, i.e., groups, classes, or clusters of a universe, in the process of problem solving.

Qualitative Spatial Reasoning: An inference mechanism that is concerned with the cognitive, computational, and formal aspects of making logical inferences by representing continuous properties of real world space by discrete systems of symbols.